The search for an optimal means of determining the minmax control parameter using sensitivity analysis

The use of computational methods for design and simulation of control systems allows for a cost-effective trial and error approach. In this work, we are concerned with the robust, real-time control of physical systems whose state space is infinite-dimensional. Such systems are known as Distributed Parameter Systems (DPS). A body whose state is heterogeneous is a distributed parameter. In particular, this work focuses on DPS systems that are governed by linear Partial Differential Equations, such as the heat equation. We specifically focus on the MinMax controller, which is regarded as being a very robust controller. The mathematical formulation of the MinMax controller involves a design parameter, &thetas;. This parameter provides a numerical measure of the robustness of the MinMax controller; hence it is very important. However, there exists no explicit formula for determining its value in advance of attempted control design. Currently, this parameter\u27s optimal value—optimal in the sense of robustness—is determined experimentally using an iterative process that seeks to maintain stability in the closed loop control system as well as an always positive definite result for [I − &thetas;2 PΠ] (i.e. [I − &thetas;2 PΠ \u3e 0) where 1 is the identity matrix, while P and Π are solutions to Algebraic Riccati Equations discussed in this dissertation. This process is obviously computationally expensive. The search for a more efficient means of determining &thetas;, including the possibility of the emergence of an explicit formula based on some mathematically rigorous criteria, is the driving force for this work. We use sensitivity analysis as a tool to mathematically investigate different criteria (such as the controller sensitivity, state sensitivity, Riccati equations\u27 sensitivity, etc.) to help achieve our goal of formulating a more efficient means of determining an optimal value for &thetas;. For each of the systems investigated, it was found that low &thetas; values (e.g. 0.05) are sufficient for adequate performance, robustness, and convergence of the MinMax controller